How do i prove how $S_5$ is generated by a two cycle and a five cycle?

How do I prove that $S_5$ (the permutation group on five letters) can be generated by a two-cycle $(12)$ and a five cycle $(12345)$?

Solution 1:

Define $s = (12345)$ and $t = (12)$. Note that $$ s\, t\, s^{-1} = (23) $$ Similarly, $$ s^2 t s^{-2} = (34), \quad s^3 t s^{-3} = (45), \quad s^4 t s^{-4} = (51) $$ From there, it's not too hard to get the rest of them. For example, $$ (13) = (23)(12)(23) $$ In this manner, we show that every transposition can be generated by $s$ and $t$. Thus, all of $S_5$ is generated by $s$ and $t$.

Lemma 2 over here should help you understand what's going on.

Solution 2:

First, verfiy the following equalities: \begin{align*} (12345)(12)(12345)^{-1} &= (23)\\ (12345)^2(12)(12345)^{-2} &= (34)\\ (12345)^3(12)(12345)^{-3} &= (45). \end{align*} You can then use these three equalities to generate all transpositions, and hence all of $S_5$.

It actually turns out that this result can be generalized. Any symmetric group $S_n$ is generated by $(12)$ and $(1, 2, \dots, n)$.